This article is about a particular curve defined by Giuseppe Peano. For other curves with similar properties, see space-filling curve.
Three iterations of a Peano curve construction, whose limit is a space-filling curve.
Two iterations of a Peano curve
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890.[1] Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve.[2]
^Peano, G. (1890), "Sur une courbe, qui remplit toute une aire plane", Mathematische Annalen, 36 (1): 157–160, doi:10.1007/BF01199438.
^Gugenheimer, Heinrich Walter (1963), Differential Geometry, Courier Dover Publications, p. 3, ISBN 9780486157207.
In geometry, the Peanocurve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano'scurve is a surjective,...
the resulting curve is the Lévy C curve. In a similar way, we can define the Koch–Peano family of curves as the set of De Rham curves generated by affine...
Giuseppe Peano (/piˈɑːnoʊ/; Italian: [dʒuˈzɛppe peˈaːno]; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of...
David Hilbert in 1891, as a variant of the space-filling Peanocurves discovered by Giuseppe Peano in 1890. Because it is space-filling, its Hausdorff dimension...
fractal Koch snowflake Boundary of the Mandelbrot set Menger sponge Peanocurve Sierpiński triangle Trees Weierstrass function The Beauty of Fractals...
B-spline Blancmange curve De Rham curve Dragon curve Koch curve Lévy C curve Sierpiński curve Space-filling curve (Peanocurve) See also List of fractals by...
generally called a curve and does not characterize sufficiently γ . {\displaystyle \gamma .} For example, the image of the Peanocurve or, more generally...
Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve and the flowsnake (a spoonerism of snowflake), is a space-filling curve whose...
number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes...
choice. Osgood curves are simple plane curves with positive Lebesgue measure (it can be obtained by small variation of the Peanocurve construction)....
known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of...
area but encloses a finite volume) Gosper curve (also known as the Peano–Gosper curve or flowsnake) Osgood curve Self-similarity Teragon Weierstrass function...
path-connected include the extended long line L* and the topologist's sine curve. However, subsets of the real line R are connected if and only if they are...
examples of space-filling curves—the Koch-Peanocurve, Cesàro and Lévy C curve, all special cases of the general de Rham curve—and following the path of...
recurrence relations in the analysis of algorithms. Space-filling curves like the Peanocurve have the same Hausdorff dimension as the space they fill. The...
these curves, each of which has a dimension D between 1 and 2 (he also mentions but does not give a construction for the space-filling Peanocurve, which...
In mathematics, the Peano surface is the graph of the two-variable function f ( x , y ) = ( 2 x 2 − y ) ( y − x 2 ) . {\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2})...
] ) = S 2 {\displaystyle \gamma ([0,1])=S^{2}} (constructed from the Peanocurve, for example), a complete proof requires more careful analysis with tools...
occurring over the whole polymer. This process follows the Space Filling PeanoCurve. It has been proposed that mammalian chromosomes form fractal globules...
is non-decreasing, and so in particular its graph defines a rectifiable curve. Scheeffer (1884) showed that the arc length of its graph is 2. Note that...
problem Patricia tree pattern pattern element P-complete PCP theorem Peanocurve Pearson's hashing perfect binary tree perfect hashing perfect k-ary tree...
all based on Douglas McKenna's space-filling curve patterns. The designs are either generalized Peanocurves, or based on a new space-filling construction...
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